Chebotarev's Density Theorem

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Chebotarev’s Density Theorem

Reinier Sorgdrager

June 28, 2020

Bachelor’s thesis Mathematics Supervisor: dr. Arno Kret

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

After deriving the class number formula, we give a proof of Chebotarev’s Density The-orem that does not invoke class field theory. We then generalize Chebotarev’s Density Theorem to the setting of an infinite Galois extension L of a number field K that is unramified except for a set of primes of K of Dirichlet density 0.

Before being able to do this we need to introduce some concepts from algebraic number theory. Most notably, we give the definition of a cycle c of a given number field and use it to define the generalized ideal class group I(c)/Pc. Then we give an asymptotic

formula for the number of integral ideals of bounded norm in a given class of I(c)/Pc.

This asymptotic formula is used to prove the class number formula, after Dirichlet series have been introduced.

Title: Chebotarev’s Density Theorem

Cover image: Eisenstein primes of norm less than 500, image by Johannes R¨ossel Author: Reinier Sorgdrager, reinier@reiniersorgdrager.com, 11870397

Supervisor: dr. Arno Kret,

Second grader: prof. dr. Lenny Taelman, End date: June 28, 2020

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1 Introduction

What is now known as Chebotarev’s Density Theorem was conjectured in 1896 by Georg Frobenius [1]. Frobenius himself only succeeded in proving a weaker version and did not live to see the proof, due to Nikolai Chebotarev, appear in the Mathematische Annalen in 1925 [2].

The theorem can be seen as a vast generalization of Dirichlet’s Theorem on Arithmetic Progressions, proven in 1837 by Gustav Lejeune Dirichlet. Dirichlet’s theorem states that for coprime positive integers m and a the amount of prime numbers occuring in the arithmetic progression

a, a + m, a + 2m, a + 3m, a + 4m, a + 5m, a + 6m, a + 7m, a + 8m, a + 9m, . . . is infinite and that the prime numbers occuring in this progression even have a density – which one can think of as the probability for a random prime number to occur in the progression – equal to 1/ϕ(m). In other words, the theorem states that the prime numbers are equidistributed over the classes in (Z /m Z)∗.

By the time Dirichlet proved this theorem, it had already been proven some decades earlier by Gauß (and conjectured by Euler in the 1770’s [3, p. 5]) that whether an integer a is a square in Fp depends only on p mod 4|a|. (This statement is essentially

equivalent to the law of quadratic reciprocity.) For example, for an odd prime p, −1 is a square modulo p if and only if p = −1 mod 4. Hence, knowing Dirichlet’s theorem, one sees that the probability that for a random prime number p there is an x ∈ Z such that −1 = x2 _{mod p is equal to 1/2.}

“What about higher powers?” one can wonder. For instance, can we compute the probability that 2 is a fifth power in Fp?

Writing Pnfor the amount of prime numbers less than n and Fnfor the amount of prime

numbers p < n for which 2 is a fifth power in Fp, we see with the aid of a computer:

n Pn Fn Fn/Pn 10 4 4 1 100 25 20 0.8 1000 168 136 0.80952380952 . . . 10, 000 1229 983 0.79983726607 . . . 100, 000 9592 7655 0.78906088407 . . . 1000, 000 78498 62793 0.79993120843 . . . 10, 000, 000 664579 531706 0.80006440167 . . . 100, 000, 000 5761455 4608953 0.79996337731 . . . 1000, 000, 000 50847534 40678259 0.80000455872 . . .

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The data suggest that the prime numbers p for which 2 is a fifth power modulo p have a density equal to 4/5.

This does not come as a surprise: an arbitrary prime number p is not congruent to 1 mod 5 with probability 3/4 and in that case the map F∗p → F∗p : x 7→ x5 will be

surjective, because 5 - # F∗p. With probability 1/4 on the other hand, p is congruent to

1 mod 5, in which case the group of fifth powers F∗,5p has index 5 in F∗p and one would

guess that 2 “lands” with probability 1/5 in the group F∗,5p . We would conclude that

the probability that 2 is a fifth power mod p is indeed 3/4 + 1/4 · 1/5 = 4/5.

However, this is merely a heuristic argument. To prove it, one needs Chebotarev’s Density Theorem. This is what we do in Chapter 7. More generally, we prove for a positive integer n and an integer a ∈ Z that the set of prime numbers p for which a is an n’th power mod p has a density equal to

1 ϕ(n) X u∈(Z /n Z)∗ 1 gcd(u − 1, n), (1.1)

under some mild assumptions on a (it may not be an n’th power in Z for example). In the case n is itself prime, this density equals (n − 1)/n, which indeed gives 4/5 in the case n = 5.

In order to give this proof, one should not look at the group (Z /n Z)∗, but at the affine group

Aff(Z /n Z) =nu t₀ ₁

: u ∈ (Z /n Z)∗, t ∈ Z /n Zo, which is the Galois group of Q(ζn, n

√

a)/ Q, where ζn is a primitive n’th root of unity.

Chebotarev’s theorem says that we obtain the density of prime numbers p for which a is an n’th power mod p, if we divide the number of so-called Frobenius elements in Aff(Z /n Z) that are associated to these primes p by the order nϕ(n) of the affine group. This is how formula 1.1 is derived.

We prove Chebotarev’s Density Theorem in Chapter 5. In the preceding chapters we develop the theory – such as that of Frobenius elements – that is required to give the proof.

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2 Preliminaries

The reader is expected to be familiar with the groups, rings, modules, and fields. The reader should know the statements covered in a first course in Galois theory. Moreover, this thesis, being in algebraic number theory, contains many algebraic number theoretic facts. All statements from algebraic number theory are either referenced or proven, but we think the reader will benefit greatly by having taken a course in algebraic number theory covering at least the (finiteness of) the class group, the splitting behaviour of primes in an extension of number fields, and Dirichlet’s Unit Theorem.

Some analysis is also needed: Chapter 3 requires knowledge of multidimensional inte-gration and Chapter 4 requires complex analysis – in both cases a first course in the subject will suffice. Finally, in Chapter 6 the reader should be familiar with the elements of measure theory and infinite Galois theory.

2.1 Rings of integers and their ideals

We define a number field K to be a field of finite degree over the rationals Q.

Let K be a number field. Then the ring of integers OK of K is defined as the integral

closure of Z in K.

The ring of integers OK has the nice property that every non-zero proper ideal a ⊂ OK

admits a unique factorization into non-zero prime ideals (which are maximal ideals), that is:

a= p1· · · pk

for certain non-zero prime ideals p1, . . . , pk and this factorization is unique up to the

order of the pi [4, ch. I, thm. 3.3].

It follows that every non-zero ideal a of OK is of finite index in the ring of integers.

In addition to the ideals of OK, we also have so-called fractional K-ideals that are defined

as non-zero finitely generated OK-submodules of K. For every fractional K-ideal a there

exists an x ∈ OK\ {0} such that a · (x) is an ideal of OK.

The fractional K-ideals turn out to form a group under ideal multiplication, with identity element OK. This group is called the group of invertible ideals and is denoted by I. A

subgroup of I is the set P of principal fractional K-ideals. The class group of K is defined as the quotient ClK = I/P .

The class group is finite and its order is the class number of K [4, ch. I, thm. 6.3].

Prime ideals in extensions of number fields

Let L/K be an extension of number fields. Then we have a corresponding extension O_K ⊂ O_L of the ring of integers. A non-zero prime ideal p ⊂ OK has an extension to

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O_L that has a unique factorization into prime ideals: pO_L= Pe1

1 · · · P ek

k ,

where the Pi are distinct prime ideals and the ei are elements of Z>0. The prime ideals

P_i are said to lie above p. If for some i we have ei > 1, then p is said to ramify in L

and we call Pi ramified over K, if this is not the case p is unramified in L and the Pi

are unramified over K.

The ei are called the ramification indices of Pi over p, and can also be denoted by

e(Pi/p) = ei.

For every i we have that

(OL/Pi)/(OK/p)

is an extension of finite fields of residue class degree f (Pi/p) := [(OL/Pi) : (OK/p)].

For a non-zero prime ideal P ⊂ OL lying above the prime ideal p ⊂ OK, we define the

degree of P over K as deg_K(P) = f (P/p). The degree deg_Q(P) is called the absolute degree of P. We have k X i=1 e(Pi/p)f (Pi/p) = [L : K],

see [4, ch. I, prop. 8.2]. The prime ideal p is said to split completely if k = [L : K], or equivalently, if e(Pi/p) = f (Pi/p) = 1 for all i.

2.2 Lattices

A lattice L ⊂ Rn is a free abelian group that is discrete as a subspace in Euclidean space. It is of the form

L = Z ·x1+ Z ·x2+ · · · + Z ·xk,

where x1, . . . , xk∈ Rnare R-linearly independent for some k called the dimension of L.

If L is n-dimensional, the parallellotope

P =n X

i

λixi : λi ∈ [0, 1)

o

is called a fundamental domain of L and it contains exactly one representant of each class in Rn/L.

In the n-dimensional case, we define the covolume of L as the volume of a fundamental domain and denote it by Covol(L). The covolume is independent of the choice of fun-damental domain, because linear isomorphisms Zn→ Zn _{have determinant ±1.}

The following theorem shows that for an n-dimensional lattice L ⊂ Znone has Covol(L) = [Zn: L].

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Theorem 2.1. Let a1, . . . , an∈ Zn be linearly independent and let P =n X i λiai : λi ∈ [0, 1) o ⊂ Rn

be the parallellotope spanned by the ai. Then

Vol(P ) = #P ∩ Zn.

Proof. Let L be the lattice generated by the ai. For a ∈ P ∩ Zn write Ca= a + [0, 1)n,

and for b ∈ L define C_ab= Ca∩ (b + P ) (for almost all b this will be the empty set).

The union S

a∈P ∩ZnCa has volume #P ∩ Zn and it has the same volume as the disjoint

union [ a∈P ∩Zn, b∈L (−b + C_ab) = P. Corollary 2.2. If α : Zn→ Zn is a Z-linear map and det(α) 6= 0, then

| det(α)| = [Zn_{: Im α].}

If L ⊂ Rn is an n-dimensional lattice and y ∈ Rn, then the translated lattice y + L does not necessarily have a group structure (that happens only if y ∈ L), but one can still think about it geometrically. This is why we define Covol(y + L) to be Covol(L).

2.3 Equivalence of norms: various definitions of the ideal

norm

Let K be a number field. Then one can define the norm of a non-zero ideal a of OK in

various ways. One such definition is as follows.

Definition (Norm of an OK-ideal). Let a be a non-zero ideal of OK. Then we define

its norm Na as

Na = #OK/a.

The norm turns out to be related to the field norm N_{K/ Q}on K, that sends an element α ∈ K to N_{L/ Q}(α) = Q

σσ(α), where the product ranges over the embeddings σ :

K → C. Note that NK/ Q(α) = det(− · α), where multiplication by α is seen as an

endomorphism of the Q-vector space K.

Lemma 2.3. The norm is multiplicative, that is, for two non-zero OK-ideals a and b

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Proof. By the Chinese Remainder Theorem it suffices to prove N(pn) = (Np)n when p is a non-zero prime ideal. We prove this by induction on n. When n = 1 it is clearly true. For n > 1 we have a surjective ring morhpism

OK/pn OK/pn−1

with kernel pn−1/pn. By the induction hypothesis we are done if we are able to show that #pn−1/pn = #OK/p. To see this, note that pn−1/pn is a non-trivial vector space

over OK/p. Its dimension cannot be greater than 1, for then we would have a proper

linear subspace of pn−1/pn, which would, when contracted to OK, give rise to an ideal

a of OK satisfying

pn_{( a ( p}n−1. In that case we would have

p ( ap1−n ( OK,

contradicting the maximality of p. Thus pn−1/pn has dimension 1 over OK/p.

Theorem 2.4. The norm Na of a non-zero ideal a ⊂ OK is equal to the unique n(a) :=

a > 0 for which a Z equals the ideal generated by the image of a under NK/ Q.

Proof. One can verify that n is multiplicative. So by multiplicativity of both N and n, it suffices to verify Np = n(p) for all non-zero prime ideals p of K.

Let h be the class number of K and let p be a non-zero prime of K. Then ph = αOK is

principal. We have

n(p)h= n(αOK) = |NK/ Q(α)| = |det(− · α)| = #OK/αOK = Nph,

where the penultimate equality follows by Corollary 2.2. Thus n(p) = Np.

2.4 Frobenius elements

Frobenius elements play a central role in this thesis. We will state and prove a few elementary results that will allow us to give their definition (cf. [5, p. 85–86]).

Throughout this section let L/K be a Galois extension of number fields with group G. Theorem 2.5. Let p ⊂ OK be a non-zero prime ideal and let Sp be the set of primes

P⊂ OLabove p. Then the G-action on Sp, given by σP := σ(P) for σ ∈ G, is transitive.

Proof. Let P be in Sp. By the Chinese Remainder Theorem there exists an x ∈ P that

is not an element of any P0 ∈ Sp different from P. Hence, for σ ∈ G, the only prime in

Sp containing σ(x) is σP. Because NL/K(x) = Q σσ(x) ∈ p ⊂ T P0_∈S pP 0 _{we conclude}

that any P0 ∈ S_p must occur as a σP.

Because G acts transitively on Spwe see that the ramification indices e(P/p) and the

residue class degrees f (P/p) are independent of P, we will hence denote them by ep and

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Corollary 2.6. For a non-zero prime p of OK one has epfpgp= [L : K].

Proof. On the one hand we have

N(pOL) = #OL/pOL= (#OK/p)[L:K],

while on the other hand N(pOL) = Y P∈Sp NPep ₌ Y P∈Sp (#OK/p)epfp = (#OK/p)epfpgp. For a prime P ∈ Sp, let GP ⊂ G be the stabilizer of P. The group GP is called the

decomposition group of P.

We have #GP = g−1p #G = epfp. Moreover GP acts naturally on OL/P, inducing a

morphism of groups GP→ Gal (OL/P)/(OK/p).

Lemma 2.7. The map GP→ Gal (OL/P)/(OK/p) is surjective.

Proof. Let x ∈ OL/P be a primitive element for (OL/P)/(OK/p) with minimal

polyno-mial f ∈ (OK/p)[X]. By the Chinese Remainder Theorem there is a lift ξ of x that is

an element of P0 for any P0 ∈ Sp\ {P}.

Let g = Q

σ∈G(X − σ(ξ)) ∈ K[X]. Its reduction modulo p looks like g = X#G−gph ∈

(OK/p)[X], for some polynomial h ∈ (OK/p)[X]. We see that the minimal polynomial

f divides h, and hence every conjugate of x over OK/p is obtained by reducing σ(ξ)

modulo P for some σ ∈ GP.

In the case that p does not ramify in L, one has ep = 1, and thus #GP = fp, which

means the map is not merely a surjection, but even an isomorphism! This gives rise to the definition of the Frobenius element of an unramified prime P ⊂ OL.

Definition (The Frobenius element of a prime P). Suppose P ⊂ OL is a non-zero

prime ideal, unramified over K, lying above the prime p ⊂ OK. Then we define the

Frobenius element (P, L/K) ∈ G of P as the unique element of GP that is sent to

(x 7→ xNp) ∈ Gal (OL/P)/(OK/p) by the map of Lemma 2.7.

Finally, note that for any non-zero prime P ⊂ OL and any σ ∈ G one has GσP =

σGPσ−1. Moreover, in the case P ⊂ OL is unramified over K (and lies above p ⊂ OK)

we have

(σP, L/K) = σ(P, L/K)σ−1, which follows from the fact that for any x ∈ OL

(σ(P, L/K)σ−1) x + σP = σ σ−1(x)Np+ P = xNp+ σP.

This means the conjugacy class of a Frobenius element of a prime P ⊂ OLdepends only

on P ∩ OK. In the case that G is abelian this conjugacy class is a singleton and we can

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3 Counting ideals

Throughout this chapter, let K be a number field and n = [K : Q].

3.1 Some more preliminaries

In this chapter, we are interested in counting the ideals of OK in a given class of a

so-called “generalized ideal class group” – whose definition will be given in the next section. The counting of ideals is an essential part in calculating the densities of prime ideals which we will do in Chapter 5 where we prove Chebotarev’s Density Theorem.

In order to count these ideals we will view them as part of Euclidean space, or more generally, we will view K as part of Euclidean space. There is one obvious way in which we can do this: using the embeddings of K into the complex numbers.

If σ : K ,→ C is an embedding, then σ induces an archimedean absolute value on K by composing the absolute value of C with σ. Note that σ and its complex conjugate σ induce the same absolute value. Conversely, one can verify that if two embeddings σ, τ : K ,→ C induce the same absolute value, then either σ = τ or σ = τ .

Hence, if K has r1 real embeddings (embeddings into R) and 2r2 complex embeddings

(embeddings into C that do not map into R), then the number of archimedean absolute values on K is r1+ r2. We will call an archimedean absolute value real if it is induced

by a real embedding and complex otherwise.

To each archimedean absolute value v on K we associate the Euclidean space Kv :=

(

R if v is real, R2 if v is complex.

If v is induced by σ : K ,→ C, then in the real case we simply get the map σ : K ,→ Kv

and in the complex case we identify C with R2 (by identifying x + iy with (x, y)) and also get a map σ : K ,→ Kv – note that we have to choose an embedding to obtain a

map K ,→ Kv in the case v is complex.

For each archimedean value v we choose an embedding σv : K ,→ Kv that induces v.

This gives us a map

ΦK : K ,→

Y

v

Kv = Rn: x 7→ (σv(x))v,

where v runs over all archimedean absolute values of K. It is this map of K into Eu-clidean n-space that we will use in the counting of ideals. When using this map we will always assume the σv have already been chosen.

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In defining the archimedean absolute value on K as above, we made sure to stress that those absolute values are the archimedean absolute values of K: for there are also non-archimedean absolute values on K. Namely, for every non-zero prime ideal p ⊂ OK

there is a p-adic absolute value on K [6, ch. 2]. We will not develop or use the theory of p-adic absolute values here, but merely mention them as a motivation for why prime ideals p and absolute values are often treated similarly (in those cases they are both referred to by “places of K”). We will see an instance of such similar treatment of absolute values and prime ideals in the definition of a “cycle” in the following section. For now, we would briefly like to mention the concept of localization at a prime ideal p. Given a non-zero prime ideal p ⊂ OK we define the localization of OK at p as

(OK)p= {a/b ∈ K : a ∈ OK, b ∈ OK\ p}.

The localization (OK)p is a local ring with maximal ideal mp = p(OK)p. We leave it to

the reader to check that the natural map OK/p → (OK)p/mp is an isomorphism.

Finally, we note that mp is a principal ideal [6, ch. I, prop. 15]. If πp∈ (OK)p generates

mp we call πp a uniformizer of p.

3.2 Generalized ideal class groups

Now we will introduce a general kind of ideal class group that can capture information about the real embeddings of a field and about ideals coprime to some given ideal. We define a cycle c of K as a formal product

c=Y

v

vm(v),

where the v run over the archimedean absolute values of K and the non-zero prime ideals of OK, and the m(v) are non-negative integers, only finitely many of which are

non-zero. (In the spirit of treating absolute values and prime ideals similarly, the letter v also denotes prime ideals here.) The m(v) can be thought of as multiplicities and just as with ideals we write v | c if m(v) > 0.

If v is an archimedean absolute value, we write v | ∞, and otherwise we write v - ∞. For each cycle c we define

c₀=Y

p-∞

pm(p).

This is called the finite part of c, and it can simply be viewed as an ideal of OK.

Let us denote the group of fractional K-ideals coprime to c (by which we mean coprime to c0) by I(c). Note that I((1)) = I, the group of fractional ideals. Just as we usually

take the quotient I/P to get the ideal class group, we will take a quotient of I(c) to obtain a generalized ideal class group. In order to do this, let us construct a suitable subgroup.

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(i) If p | c is a prime ideal, then α lies in the localization of the ring of integers at p and

α = 1 mod mm(p)p ,

where mp denotes the maximal ideal of (OK)p.

(ii) If v | c is a real absolute value induced by the embedding σ : K ,→ R, then σ(α) > 0. Denote by Kc ⊂ K∗ the subgroup of elements satisfying these conditions. (The idea

behind our notation is: given a set X we will write Xc to denote the subset of X

satisfying conditions (i) and (ii), while X(c) will denote the set of all elements of X coprime to c.) So we write Pc for the group of principal fractional ideals (α) satisfying

α ∈ Kc. We then define the generalized ideal class group of c as I(c)/Pc. If c = (1) it

equals the class group.

Like the class group, the generalized ideal class group I(c)/Pc turns out to be finite as

well. The proof presented below can be found in [6, p. 124–126]. Theorem 3.1. Let K be a number field and c = Q

vvm(v) a cycle of K. Let K(c)

denote the subset of K∗ of elements coprime to c. Then the group K(c)/Kc is finite,

and, moreover, the generalized ideal class group I(c)/Pc is finite.

Proof. Note that every class in I/P has a representative in I(c): if a ⊂ OK is an ideal

in some class A ∈ I/P we can solve the congruences α = πordpa

p mod p1+ordpa for p | c0

by the Chinese Remainder Theorem – here the πp∈ OK are uniformizers of p. Then the

ideal a · (α−1) is coprime to c and an element of A.

This shows the map I(c) → I/P is surjective, giving an isomorphism

I(c)/P (c) ∼= I/P, (3.1)

where P (c) = I(c) ∩ P denotes the group of non-zero principal ideals coprime to c. From the definitions we have Pc⊂ P (c), giving a surjective homomorphism

I(c)/Pc I(c)/P (c). (3.2)

We will now analyze its kernel P (c)/Pc.

We have a surjective group homomorphism from the elements of K∗ coprime to c to P (c), namely K(c) → P (c) : α 7→ (α). The inverse image of the group Pc is precisely

U Kc, where U = OK∗ . This induces an isomorphism

K(c)/U Kc∼= P (c)/Pc. (3.3)

If v is a real absolute value induced by the embedding σ : K ,→ R, let us write Kv+ =

R>0 ⊂ Kv. Then Kv∗/Kv+ ∼= {±1} (as we can view the Kv as fields). At last, let us

consider the map

K(c) →Y p|c0 (OK)p/mm(p)p ∗ × Y v|c v real K_v∗/K_v+,

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which maps each element of K(c) to its residue class in the corresponding component (where the images in the K_v∗/K_v+are determined by the natural maps K → Kv that we

discussed in the previous section). The kernel of this map is precisely Kc. Because the

codomain is finite, it follows that K(c)/Kc is finite. Hence K(c)/U Kc is finite as well.

By 3.3 it then follows that P (c)/Pc is finite. Because P (c)/Pc is the kernel of the map

in 3.2, it follows by 3.1 and the fact that the ideal class group is finite that I(c)/Pc is

finite as well.

If c is a cycle of K and A a class in I(c)/Pc, then A contains an OK-ideal: suppose the

fractional K-ideal a is contained in A, then for some x ∈ OK\ {0} we have a · (x) ⊂ OK.

Note that we can choose x coprime to c, so if the class of (x) in I(c)/Pc has order k, we

see that A contains the OK-ideal a · (xk).

3.3 Dirichlet’s Unit Theorem and its generalization

Recall the map ΦK : K →QvKv. Since, in constructing this map, we had to choose an

embedding σv for every complex embedding v, in a way, we threw out some information

about the other embedding. To make up for this, we will write for an archimedean absolute value v and an x ∈ Kv

kxk = (

|x| if v is real, |x|2 _{if v is complex.}

Note that for an element α ∈ K it follows that |N_{K/ Q}(α)| =Q

vkσv(α)k.

Now recall Dirichlet’s Unit Theorem.

Theorem 3.2. Let K be a number field that has r1 real absolute values and r2 complex

absolute values. Then the map L : O∗_K→Y

v

R = Rr1+r2 : x 7→ (log kσv(x)k)v,

where v ranges over all archimedean absolute values, maps the unit group O_K∗ of K to an r1+ r2− 1-dimensional lattice that lies in the hyperplane

H = {(x1, . . . , xr1+r2) ∈ R

r1+r2_: X

i

xi = 0}.

The kernel ker L is the set of roots of unity of K.

Proof. See [4, ch. I, §7], [6, p. 104–108], or [5, thm. 5.14].

The map L of Dirichlet’s Unit Theorem is often called the log map.

Just as with the ideal class group, the concept of the unit group U := O∗_K can be generalized using cycles. Let c be a cycle of K, then this “generalized unit group” is defined as Uc= U ∩ Kc. We see

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hence the index [U : Uc] ≤ [K(c) : Kc] is finite.

From this simple observation about the index of Ucin U , we obtain the following

gener-alization of Dirichlet’s Unit Theorem.

Theorem 3.3. Let K be a number field that has r1 real absolute values and r2 complex

absolute values and let c be a cycle of K. Then the map Lc: Uc→

Y

v

R = Rr1+r2 : x 7→ (log kσv(x)k)v,

where v ranges over all archimedean absolute values, maps the generalized unit group Uc

to an r1+ r2− 1-dimensional lattice that lies in the hyperplane

H = {(x1, . . . , xr1+r2) ∈ R

r1+r2_: X

i

xi = 0}.

The kernel ker Lc is the set of roots of unity that lie in Kc.

It follows that Ucmodulo its roots of unity is a free abelian group of rank r := r1+r2−1,

where r1 and r2 are defined as in the theorem above. Let u1, . . . , ur∈ O∗K be units such

that their images through Lc generate the r-dimensional lattice. Then u1, . . . , un are

called fundamental units for Uc.

Let v1, . . . , vr be r distinct archimedean values of all r + 1 archimedean absolute values

of K. Then the regulator of Uc, is defined as

Rc= det    log kσv1(u1)k · · · log kσv1(ur)k .. . . .. ... log kσvr(u1)k · · · log kσvr(ur)k    .

This definition is independent of the choice of absolute values v1, . . . , vr, since Lc maps

into the hyperplane H = {(x1, . . . , xr1+r2) ∈ R

r1+r2_: P

ixi = 0}. The determinant

is non-zero because of the linear independence of the Lc(ui). Finally, it follows from

Corollary 2.2 that Rcis independent of the choice of the ui.

In the case c = (1), the regulator Rc is called the regulator of K and is denoted by RK,

or simply by R.

3.4 Discriminants and covolumes of ideals

Let σ1, . . . , σn be the embeddings of K into C (both the real and complex embeddings).

For x1, . . . , xn∈ K we define the discriminant of x1, . . . , xn as

∆(x1, . . . , xn) =

det(σi(xj))ni,j=1

2 .

Suppose x1, . . . , xn are linearly independent over Q, then it follows from the

Artin-Dedekind Lemma about the linear independence of characters [7, thm. 12] that ∆(x1, . . . , xn) 6=

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If y1, . . . , yn∈ Q are linear independent as well, and T is the base change mapping xi to

yi, then it follows that

∆(y1, . . . , yn) = det T

2

∆(x1, . . . , xn). (3.4)

When x1, . . . , xn ∈ OK is a Z-basis for OK, we set ∆K = ∆(x1, . . . , xn), which is

independent of the choice of basis by 3.4 and Corollary 2.2. The discriminant ∆K is

called the discriminant of K.

Lemma 3.4. Let K be a number of field of degree n over Q and let a be a non-zero ideal of its ring of integers. The map ΦK : K → Rn maps a to an n-dimensional lattice that

has covolume

Covol(ΦK(a)) = 2−r2Nap|∆K|,

where r2 is the number of complex absolute values of K.

Proof. Let α1, . . . , αn be a Z-basis for a. Let σ1, . . . , σr1 be the real embeddings of K

and let τ1, . . . , τr2 be the complex embeddings that are used to define ΦK.

Write τk(α`) = xk`+iyk`. Then the discriminant of a equals the square of the determinant

of            σ1(α1) · · · σ1(αn) .. . ... x11+ iy11 · · · x1n+ iy1n .. . ... x11− iy11 · · · x1n− iy1n .. . ...            .

Adding the row corresponding to a τkto the row belonging to τk, and subsequently

sub-stracting 1₂ times the new row from the row belonging to τk shows that this determinant

equals (up to sign) that of

2r2            σ1(α1) · · · σ1(αn) .. . ... y11 · · · y1n .. . ... x11 · · · x1n .. . ...            .

The absolute value of the determinant of the latter matrix (without the scalar in front) is the volume of the parallellotope spanned by the ΦK(αi). The determinant is non-zero,

therefore a is indeed mapped to an n-dimensional lattice. By 3.4 and 2.2 it follows that

Covol(ΦK(a)) = 2−r2p|∆(α1, . . . , αn)| = 2−r2Nap|∆K|.

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3.5 Counting ideals in ideal classes

If c is a cycle of K and A a class in I(c)/Pc, then we would like to know the number

Z(t; A), which is the number of OK-ideals in A of norm ≤ t. It turns out Z(t; A) =

ρt + O(t1−1/n) as t → ∞, for some ρ ∈ R.

It is of utmost importance for Chebotarev’s Density Theorem that ρ is independent of the class A. However, the exact value of ρ is not important in order to prove the density theorem. Nevertheless, we are happy to go the extra mile and compute the exact value of ρ, because it will enable us to derive the class number formula in Theorem 4.5. The following theorem gives the value of ρ and is due to Hecke [8, thm. 121]. We will take a slightly different approach (that of Lang [6, ch. VI, §3]) than Hecke’s, because it enables us to obtain the O-term in Z(t; A) = ρt + O(t1−1/n) and it lets us deal with generalized ideal class groups rather than just the class group.

Theorem 3.5. Let K be a field of degree n over Q, let c be a cycle of K, and let A be a class of ideals in I(c)/Pc. If Z(t; A) denotes the number of (integral) ideals in A of

norm ≤ t, then Z(t; A) = 2 r1+r2_πr2_R c wc2s(c)Nc0p|∆K| t + O(t1−1/n),

where r1 is the number of real absolute values and r2 the number complex absolute values

of K, s(c) is the number of real absolute values v | c, and wc is the number of roots of

unity in Kc.

Proof. The counting of ideals can be reduced to the counting of elements of OK in some

domain D ⊂ Rn. We should construct this D with care, in order to avoid counting ideals multiple times. This is what we will do first. Afterwards it should become clear why this was the right construction.

Let Jc=ξ ∈ (R∗)r1 × (C∗)r2 : ξv > 0 for real v|c ⊂ Y v Kv, where we identify Q vKv with R n_{. Using Φ}

K we will view Kc as a subset of Jc.

Notice that the absolute value of the usual norm on K can be extended toQ

vKv: send

ξ to Nξ := Q

v|ξv|nv = Qvkξvk, where nv is 1 when v is real and 2 otherwise. (The

notation Nξ is consistent with earlier notation, for if ξ ∈ OK\ {0} then the ideal norm

N(ξ) equalsQ

vkξvk.) We define the homogenized log map

h : Jc→ Rr1+r2 : ξ 7→ log kξvk Nξnv/n v.

The image of h lies in the hyperplane H ⊂ Rr1+r2 _{of all z with}Pr1+r2

i=1 zi = 0.

Write r = r1+ r2− 1 and let V ⊂ K∗ be the free subgroup generated by fundamental

units η1, . . . , ηrof Uc(seen as subset of Jc). Writing yi= h(ηi), we see the yi’s generate an

r-dimensional lattice in H by Theorem 3.3, since the homogenized log map h restricted to Uc is simply the log map. Define F to be the fundamental domain of this lattice

consisting of the

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with 0 ≤ ci < 1. Finally define D = h−1(F ) ⊂ Jc.

Now we are ready to think about ideals.

Let b ⊂ OK be an ideal in A−1. For any ideal a ⊂ OK in A we have ab = (ξ) for some

ξ = 1 mod∗ c that is 0 mod b. So a 7→ ab is a bijection between A and the classes modulo Uc of elements ξ ⊂ OK∩ Kc satisfying ξ = 1 mod∗ cand ξ = 0 mod b.

Because F is a fundamental domain for the lattice h(V ), there is precisely one element of the h(V )-orbit of h(ξ) contained in F . Hence, adjusting for the roots of unity in Uc,

for each a ∈ A our domain D contains exactly wc such ξ representing ab.

Moreover, for arbitrary ξ in D we have the implication

(ξ = 1 mod c0∧ ξ = 0 mod b) =⇒ (ξ = 1 mod∗c∧ ξ = 0 mod b),

because D is a subset of Jc. So it is enough to consider elements in D satisfying the first

condition.

Elements ξ ∈ D satisfying ξ = 1 mod c0 and ξ = 0 mod b form a translated lattice

in Rn: every such element can be translated by an element of bc0, and by the Chinese

Remainder Theorem every such element is contained in ξ0+bc0, for any fixed ξ0 satisfying

ξ0 = 1 mod c0 and ξ0 = 0 mod b. Thus this translated lattice, which we will call L, is

simply the lattice of bc0translated by some ξ satisfying the first condition. In particular,

it has the same covolume.

Hence we find wcZ(t; A) is the same as the number of ξ ∈ L ∩ D satisfying Nξ ≤ Nb · t

(because Na ≤ t ⇐⇒ Nab ≤ Nb · t).

Notice that tD = D for t > 0: this follows from the fact that h(tx) = h(x) for all x ∈ Jc,

since ktxvk

(Ntx)nv /n =

kxvk

Nxnv /n. So if we define

D1= {x ∈ D : Nx ≤ 1},

then the elements x ∈ D of norm less than or equal to Nb · t are those contained in (Nb · t)1/nD1.

This means wcZ(t; A) equals #((Nb · t)1/nD1) ∩ L, or, equivalently,

wcZ(t; A) = #D1∩ Lt,

where we define Lt to be _(Nb·t)11/nL.

Let Pt be a fundamental domain of the lattice _(Nb·t)1 1/nbc0 and assign to every ξ ∈

D1∩ Lt the parallellotope ξ + Pt. This way we cover D1 with parallelotopes of volume

Covol(Lt) = Vol(Pt) (which tends to 0 as t → ∞). By definition of the volume, we then

find

lim

t→∞wcZ(t; A) Covol(Lt) = Vol(D1). (3.6)

Note that the volume of D1 is finite, as D1 is bounded: if x ∈ D1, we have for every

coordinate |xv| ≤ Nx1/neBr≤ eBr, where B is a bound depending on the y1, . . . , yr.

By Lemma 3.4 we have Covol(Lt) = (Nb · t)−1Covol(L) = (2r2t)−1Nc0p|∆K|. We

therefore see

Z(t; A) = 2

r2_Vol(D

1)

wcNc0p|∆K|

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Although the error term is clearly o(t), we still need to prove it is actually O(t1−1/n). If we also show that Vol(D1) = 2r1−s(c)πr2Rc, then we are done.

We will compute the volume first. Let v1, . . . , vr1+r2 be the archimedean absolute

values on K, with the v1, . . . , vr1 being real, and consider D1⊂ R

n_{in polar coordinates}

(ρi, θi) for i ∈ {1, . . . , r1+ r2}, with ρi ≥ 0 for all i,

θi=

(

1 if vi | c and i ≤ r1,

±1 otherwise if i ≤ r1,

and, for i > r1, θi∈ [0, 2π].

Recalling 3.5, we see the polar coordinates of D1 are those satisying

0 < r1+r2 Y i=1 ρni i ≤ 1 and log ρj− 1 nlog r1+r2 Y i=1 ρni i = r X i=1

cilog |σvj(ηi)|, for some ci ∈ [0, 1).

These conditions do not depend on the θi. So consider only the space P ⊂ Rr1+r2

of (ρ1, . . . ρr1+r2) satisfying these conditions. The Jacobian determinant of (ρ, θ) 7→

ρ(cos θ, sin θ) equals ρ. Thus

Vol(D1) = (2π)r22r1−s(c)

Z

P

ρr1+1· · · ρr1+r2dρ1· · · dρr1+r2.

In order to compute this integral, we will do a change of variables. Consider in the variables (u, c1, . . . , cr) the cube S = (0, 1] × [0, 1)r. Then we have a bijection f : S → P

given by ρj = fj(u, c1, . . . , cr) = u1/nexp _Xr i=1 cilog |σvj(ηi)| . In the other direction we have u = Qr1+r2

i=1 ρ ni

i , and the fact that the regulator Rc =

det(log kσj(ηi)k) r i,j=1

is non-zero ensures there is a unique solution for the ci.

Let’s compute the Jacobian determinant of f . First note ∂ρj/∂u = _n1ρj/u and ∂ρj/∂ci=

ρjlog |σvj(ηi)|. So det(Jac(f )) = 2 −r2 nρr1+1· ρr1+r2 det    1 log |σv1(η1)| · · · log |σv1(ηr)| .. . ... ... 1 log |σv_r1+r2(η1)| · · · log |σv_r1+r2(ηr)|   .

Adding the first r rows to the last after multiplying the j’th row by nj shows us

|det(Jac(f ))| = 2 −r2_R c ρr1+1· · · ρr1+r2 . Hence Vol(D1) = (2π)r22r1−s(c) Z S 2−r2_R c= 2r1−s(c)πr2Rc, as was to be shown.

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Finally, let us consider the error term in 3.7.

First note that f can be continuously extended to a map with domain [0, 1]r1+r2 _onto

the closure P of P , and by considering ef = f (un, c1, . . . , cr) we obtain a smooth map

e

f : [0, 1]r1+r2 _{→ P .}

For each real absolute value vi of K that does not divide c, the variable θi can be either

1 or −1. This means D1 has 2r1−s(c) connected components. Combining the maps ef

and θ 7→ (cos 2πθ, sin 2πθ) we obtain smooth parametrizations of the components of D1,

denoted by ef1, . . . , ef₂r1−s(c) : [0, 1]r1+r2× [0, 1]r2 = [0, 1]n→ D1, one for each component

of D1.

For the boundaries we have [

j

e

fj ∂ ([0, 1]n) ⊃ ∂D1,

which means that the boundary ∂D1 can be covered by the images of M = 2n · 2r1−s(c)

different smooth maps gj : [0, 1]n−1→ ∂D1.

When estimating the volume of D1 as in 3.6 we can only under- or overestimate near the

boundary of D1. For all t we can bound this error absolutely by Covol(Lt) · It, where It

is the number of parallellotopes intersecting the boundary: It= #

n

ξ ∈ Lt: (ξ + Pt) ∩ ∂D16= ∅

o .

So the error term in 3.7 can be bounded by It. We claim It= O(t1−1/n) as t → ∞. To

see this, cut up each side of the unit n − 1 cube [0, 1]n−1 _{into dt}1/n_{e parts of equal length}

to obtain dt1/nen−1 _{small cubes, that we collect in the set of small cubes C} t.

For a given t, the images of all small cubes in Ctthrough all maps gj cover the boundary

∂D1. The maps gj are smooth and hence Lipschitz, so the diameter of a set gj(k), k ∈ Ct

being a small cube, can be bounded by ct−1/n for some constant c independent of k, j, or t.

Finally, there is a constant S > 0, only depending on the translated lattice L1 = 1

(Nb)1/nL, such that the maximal number of parallellotopes ξ + P1 (for ξ ∈ L1) that

a set with diameter at most c can intersect is bounded by S. Thus the number of par-allellotopes ξ + Pt (for ξ ∈ Lt) that an image of a small cube gj(k) can intersect is also

bounded by S. Hence we see that we can bound Itby

M · S · dt1/nen−1= O(t1−1/n) as t → ∞, proving our last claim.

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4 Zeta functions of number fields

This chapter will essentially be the first part of chapter VIII (excluding §4) of Lang’s Algebraic Number Theory [6]. All theorems found here can be found there, with more or less the same proofs.

4.1 Dirichlet series

Dirichlet series play a central role in Chebotarev’s Density Theorem and, indeed, in number theory, as the zeta functions and L-series can be expressed as Dirichlet series. In order to study the latter functions, we will prove some elementary results about Dirichlet series. But first of all: what are they?

Definition (Dirichlet series). A Dirichlet series is a series of the form

∞

X

n=1

an/ns,

where the an are complex numbers and s is a complex variable.

If {an} and {bn} are sequences of complex numbers and An denotes the partial sum

a1+ · · · + an for n ∈ N (and A0 = 0), then recall or check the following identity of

summation by parts: n X i=m aibi= Anbn− Am−1bm+ n−1 X i=m Ai(bi− bi+1) for m ≤ n.

We are interested in when and where a Dirichlet series converges. The summation by parts identity will prove useful.

Lemma 4.1. If the Dirichlet series P an/ns converges for s = s0, then it converges

for any s with Re(s) > Re(s0), uniformly on any compact subset of this region. In

particular, it then defines an analytic function in that region. Proof. We will sum P an

ns0 1 ns−s0 by parts. Writing Pn(s0) = Pn k=1 ak

ks0 this yields for

n > m the following: n X k=m+1 ak ks0 1 ks−s0 = Pn(s0) ns−s0 − Pm(s0) (m + 1)s−s0 + n−1 X k=m+1 Pk(s0) 1 ks−s0 − 1 (k + 1)s−s0 .

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If Re(s) > Re(s0) we have 1 ks−s0 − 1 (k + 1)s−s0 = (s − s0) Z k+1 k 1 xs−s0+1dx ≤ (s − s0) k .

So if δ > 0 and Re(s) ≥ δ+Re(s0), then it follows thatPn_k=m+1ak/ks=Pn_k=m+1_kas0k _ks−s01

will get uniformly arbitrarily small, whenever |s − s0| is bounded.

Finally, this means the Dirichlet series defines an analytic series on Re(s) > Re(s0): the

series is a limit of analytic functions that converges uniformly on compact sets, hence it

is itself analytic by a theorem of Weierstraß [9, thm. III.1.3].

The previous lemma tells us the following definition – similar to the radius of conver-gence of power series – makes sense.

Definition (Abscissa of convergence). Let P an/ns be a Dirichlet series. Then the

smallest real number (or ±∞) σ0 such that the series converges for all s ∈ C with

Re(s) > σ0 is called the abscissa of convergence.

Now we will obtain a way to find an upper bound of the abscissa of convergence. Lemma 4.2. Assume here exists a C ∈ R and a σ1> 0 such that

|An| = |a1+ · · · + an| ≤ Cnσ1

for all partial sums An. Then the abscissa of convergence of P an/ns is less than or

equal to σ1.

Proof. Again writing Pn(s) =Pnk=1

ak

ks, we find for n > m by summation by parts

Pn(s) − Pm(s) = An ns − Am (m + 1)s + n−1 X k=m+1 Ak 1 ks − 1 (k + 1)s = An ns − Am (m + 1)s + n−1 X k=m+1 Aks Z k+1 k 1 xs+1dx.

Let δ > 0 and suppose Re(s) ≥ σ1+ δ. Then for all k we have

A_k Z k+1 k 1 xs+1 ≤ C Z k+1 k 1 xRe(s)−σ1+1dx, hence |P_n(s) − Pm(s)| ≤ 2C (m + 1)δ + |s|C Z n m+1 1 x1+δdx,

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Now consider

ζ(s) =X 1

ns,

which is analytic on {s ∈ C : Re(s) > 1} by the previous lemma. This function is called the Riemann zeta function and has an analytic continuation to the right half plane {s ∈ C : Re(s) > 0}, except for a single pole.

Lemma 4.3. The Riemann zeta function has an analytic continuation to {s ∈ C : Re(s) > 0}, except for a simple pole at s = 1 with residue 1.

Proof. To find the analytic continuation let us consider “the alternating zeta function” ζ2(s) = 1 − 1 2s + 1 3s − 1 4s + · · · .

By the previous lemma ζ2(s) is analytic on {s ∈ C : Re(s) > 0}, since the partial sums

of its coefficients alternate between 1 and 0 and are thus bounded. On the other hand, we have for s with Re s > 1

2 2sζ(s) + ζ2(s) = ζ(s), and hence ζ(s) = (1 − 2 2s) −1_ζ 2(s),

which gives us an analytic continuation to Re(s) > 0, except possibly for s ∈ 1 +_{log 2}2π Z i. To analyze these possible poles, consider the more general alternating zeta functions for r ∈ N: ζr(s) = 1 + 1 2s + · · · + 1 (r − 1)s − r − 1 rs + 1 (r + 1)s + · · · .

For the same reason as for r = 2 they are analytic on Re(s) > 0. And, like before, we get an identity _rrsζ(s) + ζr(s) = ζ(s) and hence another analytic continuation of the

Riemann zeta function:

ζ(s) = (1 − r rs)

−1_ζ r(s).

Taking for example ζ3(s), we see that ζ(s) can only have poles for s in 1 +_{log 3}2π Z i. So for any pole s 6= 1 we have s = 1 +_{log 2}2πin = 1 + _{log 3}2πim, implying 2n= 3m, which is only possible if m = n = 0.

Hence the only pole of ζ(s) is the one at s = 1. Note that for real s > 1 we have 1 s − 1 = Z ∞ 1 1 xsdx ≤ ζ(s) ≤ 1 + 1 s − 1. This implies 1 ≤ (s − 1)ζ(s) ≤ s for real s > 1.

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The preceding lemma is the easiest case of the class number formula, that we prove in the next section in Theorem 4.5. The theorem will in fact follow quickly from the work we have done so far. Its proof uses the following theorem along with the main theorem of the previous chapter. It is no coincidence the O-notation is turning up again!

Theorem 4.4. Let {an} be a sequence in C with partial sums An. Let 0 ≤ σ1 < 1, and

assume there is a ρ ∈ C such that

An= ρn + O(nσ1) as n → ∞.

Then the function

f (s) =

∞

X

n=1

an/ns,

defined by the Dirichlet series on {s ∈ C : Re(s) > 1 has an analytic continuation to the s ∈ C with Re(s) > σ1, except for a simple pole at s = 1 with residue ρ.

Proof. Apply Lemma 4.2 to the Dirichlet series f (s) − ρζ(s) to see it is analytic on Re(s) > σ1. Then use the previous lemma to see f (s) = (f (s) − ρζ(s)) + ρζ(s) itself is

analytic on Re(s) > σ1 except for a pole at s = 1 with residue equal to ρ.

4.2 Zeta functions and L-series

Throughout, let K be a number field and N = [K : Q].

Zeta functions

We have seen the Riemann zeta function in the previous section. One of the main reasons we are interested in it is because it allows for a vast generalization: it turns out every number field has a zeta function associated to it!

For the number field K, it is given by the (Dirichlet) series ζK(s) =

X

a

1 Nas,

where we sum over all non-zero ideals of the ring of integers of K. We call it the Dedekind zeta function of K. Note that the Dedekind zeta function of Q is the Riemann zeta function.

We are ready to state and prove the class number formula, which is an important result in its own right that gives an explicit formula for the residue of ζK(s) at s = 1. For the

purposes of proving Chebotarev’s Density Theorem, however, we note that we do not need the exact value of this residue, but only the fact that ζK(s) is analytic for s ∈ C

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Theorem 4.5 (Class number formula). The Dedekind zeta function ζK(s) is analytic

for s ∈ C with Re(s) > 1 − 1/N , except for a single simple pole at s = 1 whose residue is given by

2r1+r2_πr2_hR

wp|∆K|

,

where r1 is the number of real embeddings of K and r2 is the number of complex

em-beddings up to conjugation, h is the class number of K, R the regulator, and w is the number of roots of unity of K.

Proof. For some coefficients a1, a2, . . . we can write ζK(s) =P an/ns. The partial sum

An of the coefficients equals the number of non-zero ideals of OK of norm less than or

equal to n. Hence, by Theorem 3.5, |A_n| = X C∈I/P Z(n; C) = 2 r1+r2_πr2_hR wp|∆K| n + O(n1−1/N).

Theorem 4.4 finishes the proof.

The Dedekind zeta function has another well-known description given as a formal product, called the Euler product

Y p 1 1 − 1 Nps ,

ranging over all non-zero prime ideals.

To see it indeed describes ζK(s) we can take the logarithm of the formal product to

obtain X p − log(1 − Nps) =X m,p 1 mNpms,

using the expansion log(1 − x) =P

n−xn/n for |x| < 1.

If Re(s) = σ > 1 this sum is dominated by X m,p N mpmσ < X m,p N pmσ < N ζ(σ) < ∞,

where we sum over all rational prime numbers p.

Hence the logarithm of the product converges uniformly on {s ∈ C : Re(s) > 1} by the Weierstraß M-test, and it is thus analytic on {s ∈ C : Re(s) > 1} (again by [9, thm. III.1.3]). So when we exponentiate it again we get the product expression back, which apparently converges. Multiplying out yields

Y p 1 1 − 1 Nps =Y p 1 + 1/Nps+ 1/Np2s+ · · · =X a 1 Nas = ζK(s),

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which follows from multiplicativity of the norm map and the unique prime factorization of ideals in OK.

The logarithm of the Dedekind zeta function is of utmost importance in defining the density of sets of prime ideals in the next chapter. Especially the fact that it has a pole in s = 1 is crucial. We have X m≥2,p 1 mNpm < X p,m N p2m + X p,m N p2m+1 < ∞. Hence only X p 1 Nps

contributes to the pole of log ζK at s = 1. In fact, using a similar argument, only the

primes p of absolute degree 1 contribute to the pole: the other primes have norm equal to a perfect power of a rational prime, so the reciprocals of those norms do not contribute to the pole.

Let us write f ∼ g if f and g differ (additively) by a function that is analytic at s = 1. From the discussion above it then follows that

log ζK(s) ∼ X p 1 Nps ∼ X deg_Qp=1 1 Nps.

Because ζK(s) has a simple pole at s = 1 we have that log (s − 1)ζK is analytic around

s = 1 and thus

log 1

s − 1 ∼ log ζK(s).

Later we will gratefully use the fact that all logarithms of Dedekind zeta functions differ from each other by functions that are analytic around 1.

L-series

Another interesting class of functions are the so-called L-series. They are similar to the Dedekind zeta functions, but now we plug in some character of the ideal class group, or more generally, of I(c)/Pc for some cycle c of K. Let χ : I(c)/Pc → C∗ be such a

character. Then we define

Lc(s, χ) = Y p-c 1 −χ(p) Nps −1 ,

where by χ(p) we mean χ(p), p being the class of p in I(c)/Pc.

Just as with the zeta functions, we also have Lc(s, χ) =

X

a-c

χ(a) Nas,

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using the multiplicativity of χ.

If χ and c are trivial we obtain the Dedekind zeta function ζK(s). However, when χ is

non-trivial Lc(s, χ) has no pole at s = 1, contrary to the Dedekind zeta functions.

Theorem 4.6. If χ 6= 1 is a character of I(c)/Pc, the Dirichlet series representation for

Lc(s, χ) is convergent for Re(s) > 1 − 1/N .

Proof. Let B ∈ I(c)/Pc be such that χ(B) 6= 1, then

χ(B) X A∈I(c)/Pc χ(A) = χ(B) X A∈I(c)/Pc χ(B−1A) = X A∈I(c)/Pc χ(A),

from which we conclude thatP

A∈I(c)/Pcχ(A) = 0.

Theorem 3.5 tells us all classes in A ∈ I(c)/Pc contain, for some ρ independent of A,

ρn + O(n1−1/N) ideals of norm less than n as n → ∞. So if we write Lc(s, χ) =P an/ns,

with partial sums of the coefficients An, we obtain

An= ρ

X

A∈I(c)/Pc

χ(A) + O(n1−1/N) = O(n1−1/N) as n → ∞.

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5 Density of ideals

5.1 The Dirichlet density

From the preceding chapter we collect the following result: Lemma 5.1. For every number field K we have

log 1 s − 1 ∼ log ζK(s) ∼ X p 1 Nps ∼ X deg_Qp=1 1 Nps as s ↓ 1.

It leads to a natural definition of the density of a subset of prime ideals: the Dirichlet density.

Definition (Dirichlet density). Let K be a number field and A a subset of the set of all non-zero prime ideals of OK. Then we define the (Dirichlet) density of A as

lim s↓1 X p∈A 1 Nps log 1 s − 1 ,

if this limit exists.

A more intuitive notion of density is called “natural density” and is defined by lim

n→∞

#{p ∈ A : Np ≤ n}

#{p : Np ≤ n} .

It turns out that if the natural density exists, then the Dirichlet density exists as well and must be equal to it [10, p. 118–120]. However, the converse does not necessarily hold: there are (pathological) cases of sets of primes that have a Dirichlet density, but not a natural density [11, p. 76].

We will only work with the Dirichlet density, and shall therefore often omit the “Dirich-let” part when refering to the Dirichlet density.

Finally, we will adopt a convention common in other parts of mathematics: when all prime ideals of the ring of integers of some number field satisfy a certain property, ex-cept for those in a set of density 0, we will say that almost all prime ideals satisfy said property.

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5.2 Arithmetic progressions and cyclotomic extensions

When a and m are coprime rational integers, Dirichlet’s Theorem on Arithmetic Pro-gressions states that there are infinitely many primes in the arithmetic progression

a, a + m, a + 2m, a + 3m, a + 4m, a + 5m, . . . .

An even stronger result says the density of these primes exists and equals 1/ϕ(m). This result follows from taking K = Q in the following theorem (cf. the “Universal Norm Index Inequality” and the corollary to Theorem 8 of Chapter VIII of [6]).

Theorem 5.2. Let K(ζm)/K be a cyclotomic extension of number fields, where ζm is

an m’th primitive root of unity. Let b ∈ G = Gal(K(ζm)/K) ⊂ (Z /m Z)∗ and let Sb be

the set of unramified prime ideals p of K above which there is prime P ⊂ OK(ζm) with

Frobenius element (P, K(ζm)/K) = b. Then

Sb= {p ⊂ OK: p is unramified in K(ζm) and Np = b mod m}.

Moreover, Sb has a density, equal to

lim s↓1 X Np=b mod m 1 Nps log 1 s − 1 = 1 [K(ζm) : K] .

Proof. Let c be a cycle of K containing all real valuations, the ideal (m), and all primes of K that ramify in K(ζm) (and no other ideals). This means the prime ideals of K that

are coprime to c are unramified in K(ζm).

Let p be a prime of K that is unramified in K(ζm) and let P ⊂ OK(ζm)be a prime above

it. Looking at the residue class fields, we see the Frobenius element (P, K(ζm)/K)

(acting on OK(ζm)/P) sends ζm ∈ OK(ζm)/P to ζ

Np

m , which means (P, K(ζm)/K) =

Np ∈ (Z /m Z)∗. This proves the first statement that

Sb= {p ⊂ OK: p is unramified in K(ζm) and Np = b mod m}.

The ideals in Pc are of the form (a) with a = 1 mod∗ c, which implies a = 1 mod m

and σ(a) > 0 for every real embedding σ. We have N_{K/ Q}(a) > 0, as the norm of a is the product of its Galois conjugates and in this product, by construction, every real conjugate is positive and of course the non-real conjugates can be paired with their complex conjugates. The congruence a0 = 1 mod m also holds for the Galois conjugates a0 of a, and hence N_{K/ Q}(a) = 1 mod m. Thus N(a) = |N_{K/ Q}(a)| = 1 mod m.

As a result, the morphism I(c) → G : a 7→ (Na mod m) factors through I(c)/Pc. In this

way we can view characters of G as characters of I(c)/Pc. We will denote the set of such

characters of I(c)/Pc by X.

Let χ ∈ X be a non-trivial character. By Theorem 4.6 Lc(s, χ) is analytic around 1. We

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Write m(χ) for the order of the zero of Lc(s, χ) in s = 1. So m(χ) ≥ 0. Then for some

g analytic around 1 we have Lc(s, χ) = (s − 1)m(χ)g(s), hence

log Lc(s, χ) ∼ m(χ) log(s − 1) ∼ −m(χ) log

1 s − 1.

For s ∈ C with Re(s) > 1 and any character χ of G, we can view χ as an element of X and write log Lc(s, χ) ∼ X a∈G χ(a) X Np=a mod m 1 Nps. (5.1)

(Note that 5.1 holds even if the sum on the right hand side would range over some p | c, as there are only finitely many p dividing c.)

Summing over all characters of G, we obtain

1 −X χ6=1 m(χ) log 1 s − 1 ∼ log ζK(s) + X χ6=1 log Lc(s, χ) ∼ X χ X a∈G χ(a) X Np=a mod m 1 Nps.

Recall that the characters of G form a group (the dual group) of the same order as G, and that the elements of G can be viewed as characters of the dual group [12, ex. 5.13]. Thus X χ X a∈G χ(a) X Np=a mod m 1 Nps = #G X Np=1 mod m 1 Nps, asP

χχ(a) = 0, unless a is the identity element of G.

A prime ideal p of K splits completely in K(ζm) if and only if Np = 1 mod m, and in

that case it has precisely #G = [K(ζm) : K] primes P of K(ζm) above it. Recall only

primes P of absolute degree 1 contribute to the poles (and for these primes P ∩ K splits completely in K(ζm)). Taking all this in consideration, we find

1 −X χ6=1 m(χ) log 1 s − 1 ∼ #G X Np=1 mod m 1 Nps & X deg_QP=1 1 NPs ∼ log 1 s − 1, where we consider only real values s > 1, let s → 1, and use the sign & to mean that the right-hand side is less than or equal to the left-hand side plus some constant in a neighbourhood of 1.

This shows m(χ) = 0 for all non-trivial χ.

Now let b ∈ G and multiply both sides of (5.1) by χ(b−1). Sum over all χ and use the fact that log Lc(s, χ) is analytic around 1 for non-trivial χ to get

log 1 s − 1 ∼ log ζK(s) ∼ X a∈G X χ χ(ab−1) X Np=a mod m 1 Nps.

The sum over the χ yields 0 unless a = b mod m, and hence

log 1 s − 1 ∼ #G X Np=b mod m 1 Nps.

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This finishes the proof, as lim s↓1 X Np=b mod m 1 Nps log 1 s − 1 = 1 #G= 1 [K(ζm) : K] . Corollary 5.3 (Dirichlet’s Theorem on Arithmetic Progressions). Let m be an integer and a ∈ (Z /m Z)∗. Then the set Sa of prime numbers p ∈ Z for which a = p mod m

has a density, equal to

1 ϕ(m). . Proof. We have lim s↓1 X p=a mod m 1 ps log 1 s − 1 = lim s↓1 X N(p)=a mod m 1 Nps log 1 s − 1 = 1 [K(ζm) : K] ,

by Theorem 5.2, where the first equality holds because N(p) = p and only finitely many

p ramify in Q(ζm).

5.3 Chebotarev’s Density Theorem

Let L/K be a Galois extension of number fields and σ ∈ Gal(L/K). Write

Sσ = {p ⊂ OK: p is unramified in L and ∃P ⊂ OL above p with (P, L/K) = σ}.

In the case L/K cyclotomic we know from Theorem 5.2 that Sσ has a density equal to

1/[L : K]. For the general case it is Chebotarev’s Density Theorem that states that Sσ has a density, and the theorem also gives its value, which might be different from

1/[L : K].

Suppose p ∈ Sσ and that P ⊂ OL lies above p and (P, L/K) = σ. Then for any

τ ∈ Gal(L/K) we have (τ P, L/K) = τ στ−1. So we see that Sσ = Sτ στ−1.

Hence, supposing C ⊂ Gal(L/K) is the conjugacy class of σ, it is not be ambiguous to write SC := Sσ. We will introduce some convenient notation for the densities we are

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Definition (The (upper and lower) density of SC). For a Galois extension of number

fields L/K, a conjugacy class C ⊂ Gal(L/K), and

SC = {p ⊂ OK: p is unramified in L and ∃P ⊂ OL above p with (P, L/K) ∈ C},

we define, respectively, the upper and lower density of SC as

dsup(L/K, C) = lim sup s↓1 X p∈SC 1 Nps log 1 s − 1

and dinf(L/K, C) = lim inf s↓1 X p∈SC 1 Nps log 1 s − 1 ,

and in the case those values are equal we let

d(L/K, C) := lim s↓1 X p∈SC 1 Nps log 1 s − 1 denote the density of SC.

Now we state Chebotarev’s Density Theorem.

Theorem 5.4 (Chebotarev). Let L/K be Galois extension of number fields with group G, and let C be a conjugacy class of G. Then the density d(L/K, C) exists and equals #C/#G.

If σ is an element of C ⊂ G := Gal(L/K), then L is an abelian extension of the field of invariants Z := {x ∈ L : σx = x} with group Gal(L/Z) = hσi. The following counting argument due to Deuring [13] shows we can reduce Chebotarev’s Density Theorem to the case of an abelian extension.

Lemma 5.5 (Deuring). If d(L/Z, {σ}) = 1/[L : Z], then d(L/K, C) = #C/#G. Proof. Let SL,σ be the set of primes P of L (unramified over K) with (P, L/K) = σ.

Next, denote by S the set of primes p of K for which there is a P ∈ SL,σ dividing it (i.e.

the set whose density we want to know).

Finally, let SZbe the set of primes q of Z for which there is a P | q of L with (P, L/Z) = σ

and deg_K(q) = 1. We claim lim s↓1 X p∈S 1 Nps X q∈SZ 1 Nqs = #C[L : Z] #G . (5.2)

To see this, note that above every q ∈ SZ there lies exactly one P ∈ SL,σ: the Galois

group hσi acts transitively on the primes above q, but also σ(P) = P for such q. Con-versely, if P ∈ SL,σ it divides some non-zero prime q of Z, and as the Frobenius element

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σ of P is the identity on Z, it follows that deg_Z(P) = #hσi = deg_K(P), which implies deg_K(q) = 1, and hence q ∈ SZ. This gives a bijection between SZ and SL,σ.

For a fixed p ∈ S the number of P above p such that σ = (P, L/K) equals #Cσ/#GP,

where Cσ is the subgroup of G of elements commuting with σ and GP is the

decomposi-tion group of P. As #Cσ = #G/#C and #GP = #hσi = [L : Z], this number is equal

to

#G #C[L : Z].

For every P | p with σ = (P, L/K) there is a unique q ∈ SZ with P | q | p, moreover

Np = Nq. Hence it follows from the observations above that for any s > 1 #G #C[L : Z] X p∈S 1 Nps = X q∈SZ 1 Nqs,

and thus 5.2 is true.

Now, because only primes of Z of degree 1 over K contribute to the poles (as in fact only those of absolute degree 1 contribute to the poles), we see

1 [L : Z] = d(L/Z, {σ}) = lims↓1 X q∈SZ 1 Nqs log 1 s − 1 .

This concludes the proof, as

d(L/K, C) = lim s↓1 X p∈S 1 Nps log 1 s − 1 = lim s↓1 X p∈S 1 Nps X q∈SZ 1 Nqs · X q∈SZ 1 Nqs log 1 s − 1 = #C #G. While the original proof by Chebotarev did not, most modern proofs of his density theorem rely on class field theory. Stevenhagen and Lenstra, however, discuss a proof that does not [14]. It is this proof that is presented below.

Theorem 5.4 (Chebotarev). Let L/K be Galois extension of number fields with group G, and let C be a conjugacy class of G. Then the density d(L/K, C) exists and equals #C/#G.

Proof. Assume L 6= K, otherwise we are done. By the preceding lemma we can also assume L/K is abelian. So C = {σ} for some σ ∈ G.

For a rational prime p, let ζp be some p’th root of unity. Choose an M such that for all

p > M we have L ∩ Q(ζp) = Q.

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group of L(ζp)/K can be identified with G × H.

Now, if a prime p of K has a prime P of L(ζp) above it with Frobenius element (σ, τ ) ∈

G × H, then P ∩ L has Frobenius element σ ∈ G. Hence dinf(L/K, {σ}) ≥

X

τ ∈H

dinf(L(ζp)/K, {(σ, τ )}).

Write n := [L : K] and fix (σ, τ ) ∈ G × H. When n divides the order of τ , the subgroups h(σ, τ )i and G × {1} of G × H intersect trivially. It follows that the field of invariants F = {x ∈ L(ζp) : (σ, τ )(x) = x} satisfies F (ζp) = L(ζp), such that L(ζp)/F is cyclotomic.

Still supposing n divides the order of τ , by Theorem 5.2 we have d(L(ζp)/F, {(σ, τ )}) =

1 [L(ζp) : F ]

, and thus by Lemma 5.5

d(L(ζp)/K, {(σ, τ )}) =

1

[L(ζp) : F ][F : K]

= 1

#G#H.

Writing Hn for the set of τ ∈ H whose order is divisible by n, the above considerations

yield

dinf(L/K, {σ}) ≥

#Hn

#G#H.

Suppose k > 0 and p = 1 mod nk. Then H ∼= Z /nkr Z for some r ∈ Z>0. Suppose

n = pe1

1 · · · p e`

` with the pi distinct prime numbers and ei > 0. Then, if we consider for

each i the map

ϕi: H → H : h 7→ nkr p(k−1)ei i · h of multiplication by nkrp(1−k)ei

i , we see that # ker ϕi = nkrp (1−k)ei

i and that every

ele-ment of H whose order is not divisible by pei

i lies in ker ϕi. It follows that

#Hn #H ≥ nk_{r −}P i# ker ϕi nk_r = nk_{r −}P inkrp (1−k)ei i nk_r = 1 − X i 1 p(k−1)ei i ,

which comes arbitrarily close to 1 for large k. Dirichlet’s Theorem on Arithmetic Pro-gressions tells us there are (infinitely many) primes p > M satisfying the congruence p = 1 mod nk, for every k.

So we obtain

dinf(L/K, {σ}) ≥ 1/#G.

This is true for every σ ∈ G. We claim that dsup(L/K, {σ}) ≤ 1/#G for every σ: for

suppose this is not true for, say, σ1 ∈ G, then

1 = lim sup s↓1 X p⊂OK 1 Nps log 1 s − 1 ≥ d_sup(L/K, {σ1}) + X σ∈G\{σ1} dinf(L/K, {σ}) > 1,

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which is a contradiction.

So the upper and lower densities dsupand dinf coincide and hence the Dirichlet densities

exist for all σ ∈ G and equal

d(L/K, {σ}) = 1/#G.

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6 Chebotarev’s Density Theorem for

infinite Galois extensions

Let K be a number field and L/K a Galois extension of K with group G. Even if the extension L/K is infinite we still turn out to have a notion of Frobenius elements in G, as we will see in Section 6.2. This allows us to formulate an analogon of Chebotarev’s Density Theorem.

However, suppose that G is infinite and X ⊂ G is closed under conjugation, then the density of the primes p ⊂ OK above which there is a prime in OLwith Frobenius element

in X cannot possibly be #X/#G – as in the finite case of Chebotarev’s Density Theorem – as this expression has no meaning, because G (and possibly X) is infinite.

So in order to formulate Chebotarev’s Density Theorem in the infinite setting we need something else. Here the Haar measure comes into play: we wil be able to construct a measure h on the Galois group G with h(G) = 1, called the Haar measure. When we have this measure, we wil see that the density of the primes in OK above which there is

a prime with Frobenius in X is equal to h(X).

The following section deals with the construction of the Haar measure.

6.1 The Haar measure

We start by introducing some measure theoretic definitions which will allow us to state the rather technical result from Carath´eodory about when certain maps (pre-measures) can be extended to measures.

Definition (Semi-ring on a set). Let X be a set. Then S ⊂ P(X) is said to be a semi-ring on X if S contains the empty set, is closed under intersection, and if it holds that if S, T ∈ S, then there are finitely many disjoint S1, . . . , Sn∈ S such that S \ T =S_iSi.

Definition (Pre-measure on a semi-ring). Let X be a set and S a semi-ring on X. Then µ : S → [0, ∞) is called a pre-measure (on S) if µ(∅) = 0 and

µ [ i∈N Si = X i∈N µ(Si),

whenever S1, S2, . . . is a countable sequence of disjoint sets in S whose union is also in

S.

Carath´eodory’s Extension Theorem shows that any pre-measure on a semi-ring S on some set X can be extended to a measure on the σ-algebra generated by S (i.e. the smallest σ-algebra containing S).

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Theorem 6.1 (Carath´eodory’s Extension Theorem). Let X be a set and S a semi-ring on X. If µ : S → [0, ∞) is a pre-measure, then µ can be extended to a measure on the σ-algebra generated by S. This extension is unique if µ is defined on X and µ(X) < ∞.

Proof. See [15, thm. 6.1].

In a profinite group G there is a fundamental system of neighbourhoods of the identity consisting of open subgroups Gi of finite index in G, we will call such a system of Gi a

fundamental system of G. Because a profinite group is a topological group, the set of all a + Gi with a ∈ G forms a basis for the topology on G. Carath´eodory’s theorem allows

us to construct a Haar measure on profinite groups that have a fundamental system that is countable.

Theorem 6.2. Let G be a profinite group with countable fundamental system {Gi}i∈I,

indexed by a directed poset I. Then the map h : T (G) → [0, 1] : U 7→ sup i∈I X aGi⊂U 1 [G : Gi] ,

where T (G) is the set of opens of G, can be uniquely extended to a left-translation invariant measure on the Borel σ-algebra of G (i.e. the σ-algebra generated by the open sets).

Proof. Consider the set S = {

n

[

j=1

ajGi : a1, . . . , an∈ G, i ∈ I} ∪ {∅}

of finite unions of left-cosets of subgroups in {Gi}i∈I, containing in addition the empty

set. Note that the elements of S are compact and open, and that they form a basis for the topology of G. Define eh : S → [0, 1] by e h : n G j=1 ajGi7→ n [G : Gi] and ∅ 7→ 0. This is well-defined: if A = Fn

j=1ajGi = Fmk=1bkG` we can suppose without loss of

generality that G`⊂ Gj (because the system is directed), and we find

n [G : Gi] = X aGi⊂A 1 [G : Gi] = X aGi⊂A X bG`⊂aGi 1 [G : G`] = m [G : G`] .

This shows eh is well-defined. One can show in a similar way that eh(AtB) = eh(A)+eh(B), which means eh respects finite disjoint unions.

Suppose that F

i∈NAi ∈ S for certain A1, A2, · · · ∈ S is non-empty, then by

Chebotarev's Density Theorem